Let G be a group and H ≤G. Let x ∈G be an element of finite order n. Prove that if k is the least positive integer such that xk∈H, then k|n.
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Let G be a group and H ≤G. Let x ∈G be an element of finite order n. Prove that if k is the least positive integer such that xk∈H, then k|n.
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- Let G be a group of finite order n. Prove that an=e for all a in G.42. For an arbitrary set , the power set was defined in Section by , and addition in was defined by Prove that is a group with respect to this operation of addition. If has distinct elements, state the order of .Let G be a group and gG. Prove that if H is a Sylow p-group of G, then so is gHg1
- 24. Let be a group and its center. Prove or disprove that if is in, then and are in.43. Suppose that is a nonempty subset of a group . Prove that is a subgroup of if and only if for all and .(See Exercise 31.) Suppose G is a group that is transitive on 1,2,...,n, and let ki be the subgroup that leaves each of the elements 1,2,...,i fixed: Ki=gGg(k)=kfork=1,2,...,i For i=1,2,...,n. Prove that G=Sn if and only if HiHj for all pairs i,j such that ij and in1. A subgroup H of the group Sn is called transitive on B=1,2,....,n if for each pair i,j of elements of B there exists an element hH such that h(i)=j. Suppose G is a group that is transitive on 1,2,....,n, and let Hi be the subgroup of G that leaves i fixed: Hi=gGg(i)=i For i=1,2,...,n. Prove that G=nHi.
- True or False Label each of the following statements as either true or false. 7. If there exists an such that , where is an element of a group , then .Let be a group of order , where and are distinct prime integers. If has only one subgroup of order and only one subgroup of order , prove that is cyclic.Prove that Ca=Ca1, where Ca is the centralizer of a in the group G.
- Assume that G is a finite group, and let H be a nonempty subset of G. Prove that H is closed if and only if H is subgroup of G.Let G be a group and Z(G) its center. Prove or disprove that if ab is in Z(G), then ab=ba.Label each of the following statements as either true or false, where H is subgroup of a group G. Every group G contains at least two subgroups.